A powerful way to improve our understanding of the above scenarios is through accurate, large scale, three-dimensional numerical simulations. Nowadays, computational general relativistic astrophysics is an increasingly important field of research. In addition to the large amount of observational data by high-energy X- and -ray satellites such as Chandra, XMM-Newton, or INTEGRAL, and the new generation of gravitational wave detectors, the rapid increase in computing power through parallel supercomputers and the associated advance in software technologies is making possible large scale numerical simulations in the framework of general relativity. However, the computational astrophysicist and the numerical relativist face a daunting task. In the most general case, the equations governing the dynamics of relativistic astrophysical systems are an intricate, coupled system of time-dependent partial differential equations, comprising the (general) relativistic (magneto-)hydrodynamic (MHD) equations and the Einstein gravitational field equations. In many cases, the number of equations must be augmented to account for non-adiabatic processes, e.g., radiative transfer or sophisticated microphysics (realistic equations of state for nuclear matter, nuclear physics, magnetic fields, and so on).

Nevertheless, in some astrophysical situations of interest,
e.g., accretion of matter onto compact objects or oscillations of
relativistic stars, the ``test fluid'' approximation is enough to
get an accurate description of the underlying dynamics. In this
approximation the fluid self-gravity is neglected in comparison
to the
*background*
gravitational field. This is best exemplified in accretion
problems where the mass of the accreting fluid is usually much
smaller than the mass of the compact object. Additionally, a
description employing ideal hydrodynamics (i.e., with the
stress-energy tensor being that of a perfect fluid), is also a
fairly standard choice in numerical astrophysics.

The main purpose of this review is to summarize the existing efforts to solve numerically the equations of (ideal) general relativistic hydrodynamics. To this aim, the most important numerical schemes will be presented first in some detail. Prominence will be given to the so-called Godunov-type schemes written in conservative form. Since [163], it has been demonstrated gradually [93, 78, 244, 83, 21, 297, 229] that conservative methods exploiting the hyperbolic character of the relativistic hydrodynamic equations are optimally suited for accurate numerical integrations, even well inside the ultrarelativistic regime. The explicit knowledge of the characteristic speeds (eigenvalues) of the equations, together with the corresponding eigenvectors, provides the mathematical (and physical) framework for such integrations, by means of either exact or approximate Riemann solvers.

The article includes, furthermore, a comprehensive description of ``relevant'' numerical applications in relativistic astrophysics, including gravitational collapse, accretion onto compact objects, and hydrodynamical evolution of neutron stars. Numerical simulations of strong-field scenarios employing Newtonian gravity and hydrodynamics, as well as possible post-Newtonian extensions, have received considerable attention in the literature and will not be covered in this review, which focuses on relativistic simulations. Nevertheless, we must emphasize that most of what is known about hydrodynamics near compact objects, in particular in black hole astrophysics, has been accurately described using Newtonian models. Probably the best known example is the use of a pseudo-Newtonian potential for non-rotating black holes that mimics the existence of an event horizon at the Schwarzschild gravitational radius [217]. This has allowed accurate interpretations of observational phenomena.

The organization of this article is as follows: Section
2
presents the equations of general relativistic hydrodynamics,
summarizing the most relevant theoretical formulations that, to
some extent, have helped to drive the development of numerical
algorithms for their solution. Section
3
is mainly devoted to describing numerical schemes specifically
designed to solve nonlinear hyperbolic systems of conservation
laws. Hence, particular emphasis will be paid on conservative
high-resolution shock-capturing (HRSC) upwind methods based on
linearized Riemann solvers. Alternative schemes such as smoothed
particle hydrodynamics (SPH), (pseudo-)spectral methods, and
others will be briefly discussed as well. Section
4
summarizes a comprehensive sample of hydrodynamical simulations
in strong-field general relativistic astrophysics. Finally, in
Section
5
we provide additional technical information needed to build up
upwind HRSC schemes for the general relativistic hydrodynamics
equations. Geometrized units (*G*
=
*c*
=1) are used throughout the paper except where explicitly
indicated, as well as the metric conventions of [186]. Greek (Latin) indices run from 0 to 3 (1 to 3).

Numerical Hydrodynamics in General Relativity
José A. Font
http://www.livingreviews.org/lrr-2003-4
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